Q71. - Algebra 2

Question

Evaluate each determinant $\left|\begin{array}{ccc}1& 0& -3\\ 2& -1& 4\\ -3& 0& 2\end{array}\right|$.

Step-by-Step Solution

Answer

The determinant of$\left|\begin{array}{ccc}1& 0& -3\\ 2& -1& 4\\ -3& 0& 2\end{array}\right|=-11$ .

Step 1. Write down the given information.

The given determinant to be evaluated is$\left|\begin{array}{ccc}1& 0& -3\\ 2& -1& 4\\ -3& 0& 2\end{array}\right|$ .

Step 2. Concept used.

The determinant of$\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|=\left(a\right)\left|\begin{array}{cc}e& f\\ h& i\end{array}\right|-\left(b\right)\left|\begin{array}{cc}d& f\\ g& i\end{array}\right|+\left(c\right)\left|\begin{array}{cc}d& e\\ g& h\end{array}\right|$ .

Step 3. Calculation.

The determinant of$\left|\begin{array}{ccc}1& 0& -3\\ 2& -1& 4\\ -3& 0& 2\end{array}\right|$  is evaluated as:$\begin{array}{c}\left|\begin{array}{ccc}1& 0& -3\\ 2& -1& 4\\ -3& 0& 2\end{array}\right|=\left(1\right)\left|\begin{array}{cc}-1& 4\\ 0& 2\end{array}\right|-\left(0\right)\left|\begin{array}{cc}2& 4\\ -3& 2\end{array}\right|+(-3)\left|\begin{array}{cc}2& -1\\ 3& 0\end{array}\right|\\ =(-2-0)-\left(0\right)-3(0+3)\\ =(-2)-\left(0\right)-\left(9\right)\\ =-11\end{array}$

Step 4. Conclusion.

The determinant of$\left|\begin{array}{ccc}1& 0& -3\\ 2& -1& 4\\ -3& 0& 2\end{array}\right|=-11$ .